Determine Whether The Dilation From To Is - Chino Valley Unified School .
reduction. The distance between the vertices at (-3, 2) and (3, -2) for A is 6 and between the vertices at (-1.5, -1) and (1.5, -1) for B is 3. So the scale factor is or . 9-6 Similarity Transformations Determine whether the dilation from A to B is an enlargement or a reduction. Then find the scale factor of the dilation. 3. GAMES The dimensions of a regulation tennis court are 27 feet by 78 feet. The dimensions of a table tennis table are 152.5 centimeters by 274 centimeters. Is a table tennis table a dilation of a tennis court? If so, what is the scale factor? Explain. 1. SOLUTION: Triangle B is larger than triangle A, so the dilation is an enlargement. The distance between the vertices at (0, 0) and (4, 0) for A is 4 and between the vertices at (0, 0) and (8, 0) for B is 8. So the scale factor is SOLUTION: See if a proportion is formed using the lengths and widths of the table and the court. or 2. a table No; sample answer: Since tennis table is not a dilation of a tennis court. CCSS ARGUMENTS Verify that the dilation is a similarity transformation. 2. SOLUTION: Rectangle B is smaller than A, so the dilation is a reduction. The distance between the vertices at (-3, 2) and (3, -2) for A is 6 and between the vertices at (-1.5, -1) and (1.5, -1) for B is 3. So the scale factor is or . 3. GAMES The dimensions of a regulation tennis court are 27 feet by 78 feet. The dimensions of a table tennis table are 152.5 centimeters by 274 centimeters. Is a table tennis table a dilation of a tennis court? If so, what is the scale factor? Explain. 4. SOLUTION: Original: A(0, 0), D(–2, –2), E(2, –1) Image: A(0, 0), B(–4, –4), C(4, –2) By the Reflexive Property, . Use the distance formula to find the length of each segment: . SOLUTION: See if a proportion is formed using the lengths and widths of the table and the court. eSolutions Manual - Powered by Cognero Page 1
widths of the table and the court. and No; sample answer: Since a table 9-6 Similarity Transformations tennis table is not a dilation of a tennis court. by the Reflexive by SAS Similarity. Property, so CCSS ARGUMENTS Verify that the dilation is a similarity transformation. 5. SOLUTION: Original: J(0, 0), K(–6, 2), L(–2, 6) Image: J(0, 0), R(–3, 1), S(–1, 3) 4. SOLUTION: Original: A(0, 0), D(–2, –2), E(2, –1) Image: A(0, 0), B(–4, –4), C(4, –2) By the Reflexive Property, By the Reflexive Property, Use the distance formula to find the length of each segment: . . Use the distance formula to find the length of each segment: . and and Property, so by the Reflexive Property, so by SAS Similarity. by the Reflexive by SAS Similarity. Determine whether the dilation from A to B is an enlargement or a reduction. Then find the scale factor of the dilation. eSolutions Manual - Powered by Cognero Page 2 5. SOLUTION: . 6.
The distance between the vertices at (0, 0) and (2.5, 1) for B is and by the Reflexive 9-6 Similarity Transformations Property, so by SAS Similarity. So, the scale factor is . Determine whether the dilation from A to B is an enlargement or a reduction. Then find the scale factor of the dilation. 8. SOLUTION: Parallelogram B is smaller than parallelogram A, so the dilation is a reduction. 6. SOLUTION: Kite B is larger than kite A, so the dilation is an enlargement. The distance between the vertices at (0, 0) and (-6, 0) for A is 6 and between the vertices at (0, 0) and (2, 0) for B is 2. The distance between the vertices at (0, 0) and (0, 4) for B is 4-0 4 units and between the vertices at (0, So the scale factor is or . 0) and (0, ) for A is . So, the scale factor is . 9. SOLUTION: Trapezoid B is larger than trapezoid A, so the dilation is an enlargement. 7. The distance between the vertices at (0, 0) and (0, 6) for B is 6 and between the vertices at (0, 0) and SOLUTION: Triangle B is smaller than triangle A, so the dilation is a reduction. (0, -3) for A is 3. So the scale factor is The distance between the vertices at (0, 0) and (5, 2) for A is or 2. Determine whether each dilation is an enlargement or reduction. The distance between the vertices at (0, 0) and (2.5, 1) for B is So, the scale factor is . 10. SOLUTION: Since the "after" image of the pupil is larger than the "before" image, this is an enlargement. 11. Refer to Page 513. eSolutions Manual - Powered by Cognero 8. SOLUTION: SOLUTION: This is a reduction because the postcard is a smaller Page 3 and similar version of the painting. YEARBOOK
10. SOLUTION: Since the "after" image of the pupil is larger than the 9-6 Similarity Transformations "before" image, this is an enlargement. 11. Refer to Page 513. SOLUTION: This is a reduction because the postcard is a smaller and similar version of the painting. No; sample answer: Since the design and the actual tattoo are not proportional. Therefore, the tattoo is not a dilation of the design. Graph the original figure and its dilated image. Then verify that the dilation is a similarity transformation. 14. M (1, 4), P(2, 2), Q(5, 5); S(–3, 6), T(0, 0), U(9, 9) SOLUTION: 12. YEARBOOK Jordan is putting a photo of the lacrosse team in a full-page layout in the yearbook. The original photo is 4 inches by 6 inches. If the photo in the yearbook is inches by 10 inches, is the yearbook photo a dilation of the original photo? If so, what is the scale factor? Explain. SOLUTION: Yes; sample answer: The dimensions of the photo are 4 inches by 6 inches, the ratio of the sides is The dimensions of the yearbook are Use the distance formula to find the lengths of the sides. by 10 inches, which results in a ratio of The photo in the yearbook is a dilation of the original photo. The scale factor is 13. CCSS MODELING Candace created a design to be made into temporary tattoos for a homecoming game as shown. Is the temporary tattoo a dilation of the original design? If so, what is the scale factor? Explain. SOLUTION: See if a proportion is formed using the lengths and widths of the design and tattoo. No; sample answer: Since the design and the actual tattoo are not proportional. Therefore, the tattoo is not a dilation of the design. Graph the original figure and its dilated image. Then verify that the dilation is a similarity transformation. 14. M (1, 4), P(2, 2), Q(5, 5); S(–3, 6), T(0, 0), U(9, 9) SOLUTION: eSolutions Manual - Powered by Cognero so by SSS Similarity. 15. A(1, 3), B(–1, 2), C(1, 1); D(–7, –1), E(1, –5) SOLUTION: Page 4
By the Reflexive Property so 9-6 Similarity Similarity. Transformations by SSS so and by SAS Similarity. 15. A(1, 3), B(–1, 2), C(1, 1); D(–7, –1), E(1, –5) 16. V(–3, 4), W(–5, 0), X(1, 2); Y(–6, –2), Z(3, 1) SOLUTION: SOLUTION: Use the distance formula to find the lengths of the sides. Use the distance formula to find the lengths of the sides. By the Reflexive Property so and by SAS Similarity. 16. V(–3, 4), W(–5, 0), X(1, 2); Y(–6, –2), Z(3, 1) SOLUTION: and Property, so by the Reflexive by SAS Similarity. 17. J(–6, 8), K(6, 6), L(–2, 4); D(–12, 16), G(12, 12), H (–4, 8) SOLUTION: Use the distance formula to find the lengths of the sides. eSolutions Manual - Powered by Cognero Use the distance formula to find the lengths of the sides. Page 5
and by the Reflexive 9-6 Similarity Transformations Property, so by SAS Similarity. so by SSS Similarity. 17. J(–6, 8), K(6, 6), L(–2, 4); D(–12, 16), G(12, 12), H (–4, 8) If find the missing coordinate. SOLUTION: 18. SOLUTION: Use the distance formula to find the lengths of the sides. Since . AB 4, AY 8, and AC 6 Substitute. The coordinates of Z are (12, 0) because the point Z lies on the right side of the x-axis, A lies on the origin, and AZ 12. 19. SOLUTION: Since . AZ 12, AY 6, and AC 4 Substitute. Since the point B lies below the y axis, the coordinates of B are (0, –2), A lies on the origin, and AB 2. so by SSS 20. GRAPHIC ART Aimee painted the sample sign Similarity. shown using If find the missing coordinate. eSolutions Manual - Powered by Cognero bottle of glass paint. The actual sign she will paint in a shop window is to be 3 feet by Page 6 feet.
Since the point B lies below the y axis, the coordinatesTransformations of B are (0, –2), A lies on the origin, and 9-6 Similarity AB 2. b. GEOMETRIC Repeat the process in part a two times. Label the second pair of triangles MNP and MQR and the third pair TWX and TYZ. Use different scale factors than part a. c. TABULAR Complete the table below with the appropriate values. 20. GRAPHIC ART Aimee painted the sample sign shown using bottle of glass paint. The actual sign she will paint in a shop window is to be 3 feet by feet. a. Explain why the actual sign is a dilation of her sample. b. How many bottles of paint will Aimee need to complete the actual sign? SOLUTION: a. Since , the new sign is a dilation of her sample. d. VERBAL Make a conjecture about how you could predict the coordinates of a dilated triangle with a scale factor of n if the two similar triangles share a corresponding vertex at the origin. SOLUTION: a. When drawing your two triangles, where one is twice as large as the other, you will need to make each side length of the new triangle ( ) twice as long as the smaller triangle ( ). b. For 6 inches by 15 inches sign, or 0.5 foot by 1.25 feet sign, she used ½ bottle of glass paint. Therefore, Aimee needs 36(0.5) bottles or 18 bottles of paint to complete the actual sign. 21. MULTIPLE REPRESENTATIONS In this problem, you will investigate similarity of triangles on the coordinate plane. a. GEOMETRIC Draw a triangle with vertex A at the origin. Make sure that the two additional vertices B and C have whole-number coordinates. Draw a similar triangle that is twice as large as with its vertex also located at the origin. Label the triangle ADE. b. GEOMETRIC Repeat the process in part a two times. Label the second pair of triangles MNP and MQR and the third pair TWX and TYZ. Use different scale factors than part a. c. TABULAR Complete the table below with the appropriate values. eSolutions Manual - Powered by Cognero d. VERBAL Make a conjecture about how you b. When drawing your two triangles, where one is twice as large as the other, you will need to make each side length of the new triangle twice as long as the smaller triangle. Try making figures in different quadrants to obtain a variety of coordinates. Page 7
b. When drawing your two triangles, where one is twice as large as the other, you will need to make each side length of the new triangle twice as long as the smallerTransformations triangle. Try making figures in different 9-6 Similarity quadrants to obtain a variety of coordinates. d. Sample answer: Multiply the coordinates of the given triangle by the scale factor to get the coordinates of the dilated triangle. 22. CHALLENGE MNOP is a dilation of ABCD. How is the scale factor of the dilation related to the similarity ratio of ABCD to MNOP? Explain your reasoning. SOLUTION: Reciprocals; sample answer: The similarity ratio of ABCD to MNOP can be expressed using the ratio The scale factor is the ratio Therefore, the similarity ratio of ABCD to MNOP and the scale factor are reciprocals. For example, in the similar parallelograms ABCD and MNOP below, the similarity ratio of the corresponding sides is . Since MNOP is larger than ABCD, it is an enlargement and the scale factor is . c. Carefully record the coordinates of your triangles in the provided table. Pay attention to any patterns noticed when comparing the coordinates of the small triangle and the enlarged triangle. 23. CCSS REASONING The coordinates of two triangles are provided in the table. Is a dilation of Explain. d. Sample answer: Multiply the coordinates of the given triangle by the scale factor to get the coordinates of the dilated triangle. 22. CHALLENGE MNOP is a dilation of ABCD. How is the scale factor of the dilation related to the similarity ratio of ABCD to MNOP? Explain your reasoning. SOLUTION: Reciprocals; sample answer: The similarity ratio of ABCD to MNOP can be expressed using the ratio The scale factor is the ratio Therefore, the similarity ratio of ABCD to MNOP and the scale factor are reciprocals. For example, in the similar parallelograms ABCD and MNOP below, the similarity ratio of the corresponding sides is . Since MNOP is larger than ABCD, it is an enlargement and the scale eSolutions Manual - Powered by Cognero factor is . SOLUTION: No; sample answer: For one triangle to be a dilation of the other, their scale factor must enlarge or reduce the transformation proportionally. Since the xcoordinates are multiplied by 3 and the y-coordinates are multiplied by 2, is 3 times as wide and only 2 times as tall as . Therefore, the transformation is not a dilation. OPEN ENDED Describe a real-world example of each transformation other than those given in this lesson. 24. enlargement SOLUTION: Choose a real-world example in which a new image Page 8 created is proportionally larger than the original. Sample answer: An image formed using a digital projector is an enlargement.
the transformation proportionally. Since the xcoordinates are multiplied by 3 and the y-coordinates are multiplied by 2, is 3 times as wide and only 2 times as tall as 9-6 Similarity Transformations . Therefore, the transformation is not a dilation. OPEN ENDED Describe a real-world example of each transformation other than those given in this lesson. 24. enlargement SOLUTION: Choose a real-world example in which a new image created is proportionally larger than the original. Sample answer: An image formed using a digital projector is an enlargement. 25. reduction SOLUTION: Choose a real-world example in which a new image created is exactly the same size and dimensions as the original image. Sample answer: Stamps are congruence transformations. 27. WRITING IN MATH Explain how you can use scale factor to determine whether a transformation is an enlargement, a reduction, or a congruence transformation. SOLUTION: Sample answer: If a transformation is an enlargement, the lengths of the transformed object will be greater than the original object, so the scale factor will be greater than 1. In the example below, is an enlargement of with a scale factor of . SOLUTION: Choose a real-world example in which a new image created is proportionally smaller than the original. Sample answer: Architectural plans are reductions. 26. congruence transformation SOLUTION: Choose a real-world example in which a new image created is exactly the same size and dimensions as the original image. Sample answer: Stamps are congruence transformations. 27. WRITING IN MATH Explain how you can use scale factor to determine whether a transformation is an enlargement, a reduction, or a congruence transformation. SOLUTION: Sample answer: If a transformation is an enlargement, the lengths of the transformed object will be greater than the original object, so the scale factor will be greater than 1. In the example below, is an enlargement of with a scale factor of . If a transformation is a reduction, the lengths of the transformed object will be less than the original object, so the scale factor will be less than 1, but greater than 0. In the example below, is a reduction of If the transformation is a congruence transformation, the scale factor is 1, because the lengths of the transformed object are equal to the lengths of the original object.In the example below, is congruent to If a transformation is a reduction, the lengths of the transformed object will be less than the original object, so the scale factor will be less than 1, but eSolutions Manual byexample Cognero below, greater than- Powered 0. In the is a reduction of with a scale factor of . with a scale factor of . with a scale factor of . 28. ALGEBRA Which equation describes the line that Page 9 passes through (–3, 4) and is perpendicular to 3x – y 6?
The length of a side of square B is 9-6 Similarity Transformations the length of a side of square A. So, the scale factor is 28. ALGEBRA Which equation describes the line that passes through (–3, 4) and is perpendicular to 3x – y 6? . 30. In the figure below, A C. A B C y 3x 4 D y 3x 3 SOLUTION: The slope of the line 3x – y 6 is 3. So, the slope of the perpendicular line of 3x – y 6 is . Use the point-slope formula. Which additional information would not be enough to prove that ADB CEB? F. G. H. J. ADB CEB SOLUTION: If So, the correct option is B. 29. SHORT RESPONSE What is the scale factor of the dilation shown below? A C and The length of a side of square B is the length of a side of square A. So, the scale factor is . 30. In the figure below, A C. eSolutions Manual - Powered by Cognero Which additional information would not be enough to prove that ADB CEB? ADB CEB by SAS Similarity. If A C and ADB CEB, then ADB CEB by AA Similarity. If , then ABD and CBE are both right angles. Since all right angles have a measure of 90, then ABD CBE. If A C and ABD CBE, then ADB CEB by AA Similarity. If A C and , then there is not enough information to prove that ADB CEB. The correct answer is H. 31. SAT/ACT If SOLUTION: , then then y A 4 B 2 C1 D E SOLUTION: Page 10
ABD CBE, then ADB CEB by AA Similarity. If A C and , then there is not enough information to prove that ADB CEB. 9-6 Similarity Transformations The correct answer is H. 31. SAT/ACT If then y So, the length of the longest side of the second garden is 15 feet. Determine whether answer. Justify your A 4 B 2 C1 D E 33. AC 8.4, BD 6.3, DE 4.5, and CE 6 SOLUTION: SOLUTION: yes; 34. AC 7, BD 10.5, BE 22.5, and AE 15 SOLUTION: yes; So, the correct choice is E. 32. LANDSCAPING Shea is designing two gardens shaped like similar triangles. One garden has a perimeter of 53.5 feet, and the longest side is 25 feet. She wants the second garden to have a perimeter of 32.1 feet. Find the length of the longest side of this garden. SOLUTION: If two triangles are similar, the lengths of corresponding medians are proportional to the lengths of corresponding sides. Form a proportion for the given situation. Let x be the length of the longest side of the second garden. 35. AB 8, AE 9, CD 4, and CE 4 SOLUTION: no; If each figure is a kite, find each measure. 36. QR SOLUTION: By the Pythagorean Theorem, 2 2 2 QR 8 6 100 So, the length of the longest side of the second garden is 15 feet. Determine whether answer. Justify your Since the length must be positive, QR 10. 37. m K 33. AC 8.4, BD 6.3, DE 4.5, and CE 6 eSolutions Manual - Powered by Cognero SOLUTION: Page 11 SOLUTION: A kite can only have one pair of opposite congruent
By the Pythagorean Theorem, 2 2 2 2 2 DC 12 5 169 2 QR 8 6 100 9-6 Similarity Transformations Since the length must be positive, QR 10. 37. m K SOLUTION: A kite can only have one pair of opposite congruent angles and . Let The sum of the measures of the angles of a quadrilateral is 360. Since the length must be positive, DC 13. Here, DC BC. So, BC 13. 39. PROOF Write a coordinate proof for the following statement. If a line segment joins the midpoints of two sides of a triangle, then it is parallel to the third side. SOLUTION: The "if" statement contains the given information for your proof; namely that you know the midpoints of two sides of a triangle. The "then" part of the sentence contains what you are trying to prove; that the midsegment formed by the midpoints of two sides is parallel to the third side of the triangle. It is helpful to place one of the sides of the triangle on the x-axis, for ease of calculation of coordinates and midpoints. Create a triangle and label the vertices A(0,0), B( a, 0), and C ( b, c).Calculate the midpoint of each side, in terms of a, b, and c. Then, compare the slope of the midsegment and the third side of the triangle, to prove that they are parallel. So, 38. BC SOLUTION: By the Pythagorean Theorem, 2 2 2 DC 12 5 169 Since the length must be positive, DC 13. Here, DC BC. So, BC 13. Given: S is the midpoint of T is the midpoint of Prove: Proof: Midpoint S is or Midpoint T is 39. PROOF Write a coordinate proof for the following statement. If a line segment joins the midpoints of two sides of a triangle, then it is parallel to the third side. SOLUTION: The "if" statement contains the given information for your proof; namely that you know the midpoints of two sides of a triangle. The "then" part of the sentence contains what you are trying to prove; that the midsegment formed by the midpoints of two sides is parallel to the third side of the triangle. It is helpful to place one of the sides of the triangle on the x-axis, for ease of calculation of coordinates and midpoints. eSolutions Manual - Powered by label Cognero Create a triangle and the vertices A(0,0), B( a, 0), and C ( b, c).Calculate the midpoint of each side, in terms of a, b, and c. Then, compare the slope of Slope of Slope of have the same slope so Solve each equation. 40. 145 29 · t SOLUTION: Page 12
Slope of 9-6 Similarity Transformations have the same slope so Solve each equation. 40. 145 29 · t SOLUTION: . 45. SOLUTION: 41. 216 d · 27 SOLUTION: 42. 2r 67 · 5 SOLUTION: 43. SOLUTION: 44. SOLUTION: . 45. eSolutions Manual - Powered by Cognero SOLUTION: Page 13
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Chính Văn.- Còn đức Thế tôn thì tuệ giác cực kỳ trong sạch 8: hiện hành bất nhị 9, đạt đến vô tướng 10, đứng vào chỗ đứng của các đức Thế tôn 11, thể hiện tính bình đẳng của các Ngài, đến chỗ không còn chướng ngại 12, giáo pháp không thể khuynh đảo, tâm thức không bị cản trở, cái được
Le genou de Lucy. Odile Jacob. 1999. Coppens Y. Pré-textes. L’homme préhistorique en morceaux. Eds Odile Jacob. 2011. Costentin J., Delaveau P. Café, thé, chocolat, les bons effets sur le cerveau et pour le corps. Editions Odile Jacob. 2010. Crawford M., Marsh D. The driving force : food in human evolution and the future.
Le genou de Lucy. Odile Jacob. 1999. Coppens Y. Pré-textes. L’homme préhistorique en morceaux. Eds Odile Jacob. 2011. Costentin J., Delaveau P. Café, thé, chocolat, les bons effets sur le cerveau et pour le corps. Editions Odile Jacob. 2010. 3 Crawford M., Marsh D. The driving force : food in human evolution and the future.
